Efficient D-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems

Abstract

We develop a computational framework for D-optimal experimental design for PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We follow a formulation of the experimental design problem that remains valid in the infinite-dimensional limit. The optimal design is obtained by solving an optimization problem that involves repeated evaluation of the log-determinant of high-dimensional operators along with their derivatives. Forming and manipulating these operators is computationally prohibitive for large-scale problems. Our methods exploit the low-rank structure in the inverse problem in three different ways, yielding efficient algorithms. Specifically, we propose three approaches for computing the D-optimal criterion, its gradient, and the Kullback--Leibler (KL) divergence from the posterior to prior. The first approach is based on truncated spectral decomposition of the prior-preconditioned data misfit Hessian, the second approach uses randomized matrix methods, and the third approach uses a fixed low-rank approximation of the prior-preconditioned forward operator. Detailed error analysis is provided for each of the methods, and their effectiveness is demonstrated on a model sensor placement problem for initial state reconstruction in a time-dependent advection-diffusion equation in two space dimensions.

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